Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A generalization of the Auslander-Nagata
purity theorem

Author: Miriam Ruth Kantorovitz
Journal: Proc. Amer. Math. Soc. 127 (1999), 71-78
MSC (1991): Primary 13B15; Secondary 13B02
MathSciNet review: 1458881
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $B \hookrightarrow A$ be a module finite extension of normal domains. We show that if $B \hookrightarrow A$ is unramified in codimension one and if $A$ has finite projective dimension over $B$, then $A$ is étale over $B$. Our proof makes use of P. Roberts' New Intersection Theorem.

References [Enhancements On Off] (What's this?)

  • [Ab] S. Abhyankar, in Ramification Theoretic Methods in Algebraic Geometry, Annals of Mathematics Studies Vol. 43, Princeton: Princeton Univ. Press, 1959.MR 21:4158
  • [Au] M. Auslander, On the purity of branch locus, Amer. J. Math. 84 (1962), 116-125. MR 25:1182
  • [AB] M. Auslander and D. Buchsbaum, On ramification theory in Noetherian rings, Amer. J. Math. 81 (1959), 749-765. MR 21:5659
  • [Bor] A. Borek, Weak purity for Gorenstein rings, J. of Alg. 175 (1995), 409-450. MR 96i:13010
  • [Bou] N. Bourbaki, Commutative Algebra, Chapters 1-7, Berlin, Heidelberg, New York: Springer-Verlag, 1989. MR 90a:13001
  • [BH] W. Bruns and J. Herzog, Cohen Macaulay Rings, Cambridge: Cambridge University Press, 1993. MR 95h:13020
  • [Ch] W. L. Chow, On the theorem of Bertini for local domains, Proceedings of the National Academy of Science, USA, 44 (1958), 580-584. MR 20:3150
  • [Cu] S. D. Cutkosky, Purity of the branch locus and Lefschetz theorems, Compositio Math. 96, No 2 (1995), 173-195. MR 96h:13023
  • [F] R. Fossum, The Divisor Class Group of Krull Domains, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol 74, Berlin, Heidelberg, New York: Springer-Verlag, 1973. MR 52:3139
  • [G1] P. Griffith, Normal extensions of regular local rings, J. of Alg. 106 (1987), 465-475. MR 88c:13020
  • [G2] P. Griffith, Some results in local rings in ramification in low codimension, J. of Alg. 137 (1991), 473-490. MR 92c:13017
  • [SGA2] A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux in Séminaire de Géometrie Algébrique (SGA), fasc. I and fasc. II, IHES, 1962, 1963. MR 57:16294; MR 35:1604
  • [HW1] C. Huneke and R. Wiegand, Tensor products of modules and the rigidity of Tor, Math. Ann. 299 (1994), 449-476. MR 95m:13008
  • [HW2] C. Huneke and R. Wiegand, Tensor products of modules, rigidity, and local cohomology, submitted.
  • [Ma] H. Matsumura, Commutative Ring Theory, Cambridge: Cambridge University Press, 1989. MR 90i:13001
  • [Mi] C. Miller, Hypersurface sections: A study of divisor class groups and the complexity of tensor products, Ph. D. thesis, University of Illinois at Urbana-Champaign, 1996.
  • [N1] M. Nagata, On the purity of branch locus in regular local rings, Ill. Jour. of Math. 3 (1959), 328-333. MR 21:5660
  • [N2] M. Nagata, Local Rings, Interscience Tracts in Pure and Applied Mathematics, Vol. 13, New York: Wiley, 1962. MR 27:5790
  • [R] P. Roberts, Le théorème d'intersection, C. R. Acad. Sc. Paris Sér. I 304 (1987), 177-180. MR 89b:14008
  • [W] K. Watanabe, Certain invariant subrings are Gorenstein I, Osaka J. Math. 11 (1974), 1-8. MR 50:7124
  • [Z] O. Zariski, On the purity of branch locus of algebraic functions, Proc. Nat. Acad. U.S.A. 44 (1958), 791-796. MR 20:2344

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 13B15, 13B02

Retrieve articles in all journals with MSC (1991): 13B15, 13B02

Additional Information

Miriam Ruth Kantorovitz
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

Keywords: Auslander-Nagata purity, unramified extension
Received by editor(s): October 17, 1996
Received by editor(s) in revised form: May 14, 1997
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society